Wellposedness and regularity of the variable-order time-fractional diffusion equations

Analysis and efficient implementation of alternating direction implicit finite volume method for Riesz space‐fractional diffusion equations in two space dimensions

“…As have been pointed out in [6, 26] that the solutions to FPDEs exhibit nonphysical singularities near the boundary of domain (or the initial time t = 0), which means the regularity assumption of solutions in Theorem 3.9 may not fulfilled that in turn affects the optimal convergence rates of the numerical methods [9]. Recently, it was demonstrated in [30] that the variable‐order FPDEs where the order of the fractional derivatives has an integer limit near the boundary of domain (or the initial time t = 0) are able to eliminate the nonphysical singularity of the solutions to constant‐order FPDEs. This observation shows the strong potential of variable‐order FPDEs for modeling the multiphysical phenomena.…”

Uniqueness of determining the variable fractional order in variable-order time-fractional diffusion equations

“…Then the following theorems hold [32]. If α (l) (0) = 0 for 0 ≤ l ≤ n − 2 and lim t→0 α (n−2) (t) ln t exists, then u ∈ C n ([0, T ];Ȟ γ (0, L)) such that…”

Optimal-order error estimates of finite element approximations to variable-order time-fractional diffusion equations without regularity assumptions of the true solutions

“…Note thatȞ γ (Ω) is a subspace of the fractional Sobolev space H γ (Ω) characterized by [1,17,20] H γ (Ω) = v ∈ H γ (Ω) : L s v(x) = 0, x ∈ ∂Ω, s < γ/2 and the seminorms |v|Ȟγ and |v| H γ are equivalent inȞ γ . Then using the approaches of variable seperation, the wellposedness of the model (1) and the regularity estimates of its solutions u(x) are proved by the following theorems [24]. Here Q = Q αm, k C 1 [0,T ] , T .…”